The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that distinct pretopologies can give rise to the same topology on a category. My (almost certainly trivial) questions are the following:

1) What are the key differences between Grothendieck topologies and Grothendieck pretopologies, and

2) What are some nice examples of distinct pretopologies that give rise to the same topology?

4 Answers
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A) Let $\tau$ be a Grothendieck pretopology on any category with fiber products. Define a new Grothendieck pretopology $\tau'$ where a cover $\{U_i \to X\}$ is a $\tau'$ cover if and only if there exists a refinement $\{V_{ij} \to X\}$ (i.e. there exists for each $ij$ an $X$-morphism $V_{ij} \to U_i$) such that $\{V_{ij} \to X\}$ is a $\tau$ cover. Then $\tau$ and $\tau'$ generate the same topologies.

B) The operation (lets say $\Phi$) that takes topologies to pretopologies composed with the operation (say $\Psi$) that takes pretopologies to topologies gives back the same topology that you started with (i.e. $\Psi \Phi = id$). So for any pretopology $\tau$, the pretopology $\Phi\Psi \tau$ gives the same topology as $\tau$.

C) Related to (A), on the category of affine schemes, we can define an fppf cover as a family $\{U_i \to X\}$ which is jointly surjective, and such that each morphism is flat, and finitely presented. We could also include the requirement that in addition the morphisms be quasi-finite. Corollary 17.16.2 in EGA IV implies that these two pretopologies gives rise to the same topology.

In response to the first question, there are surely people more expert than me, but it seems to me that Grothendieck topologies in algebraic geometry are almost always defined via Grothendieck pretopologies. I think topologies are the nicer concept, but while the idea is suited to proving very general things about topoi, if you have a specific pretopology such as the étale, Zariski, Nisnevich, flat, cdh, envelopes (also called proper cdh - see 18.3 in Fulton's "Intersection theory" and the Mazza, Voevodsky, Weibel book), etc having actual scheme morphisms in your hands allows you to apply the strong algebr-geometric results which are largely responsible for making topologies such a powerful tool in algebraic geometry. Of course, I'm sure someone in logic would have a very different point of view.

In relation to this line of thought, Voevodsky - while working with the Nisnevich and cdh topologies - found it useful to further simplify the data leading to a topology in the notion of a cd structure (see the papers "Unstable motivic homotopy categories in
Nisnevich and cdh-topologies" and "Homotopy theory of simplicial sheaves in completely decomposable topologies").

While I'm talking about Voevodsky's work, he also uses from time to time an idea which he calls "covers of normal form" in "Homology of schemes I", but the same idea is used implicitely in the appendix to "Singular homology of abstract algebraic varieties". The way I understand this phenomenon is the following. If you have two pretopologies $\sigma$ and $\rho$ on a category with fiber products, then the covers of the pretopology generated by $\sigma$ and $\rho$ are finite compositions of covers which are either a $\sigma$ cover or a $\rho$ cover. Lets denote the new pretopology by $\langle \sigma, \rho \rangle$. Many pretopologies $\tau$ in common use, are actually generated by two other pretopologies in the sense that $\langle \sigma, \rho \rangle$ and $\tau$ give rise to the same topology.

For example, the cdh pretopology is generated in this way (by definition) by the Nisnevich pretopology and the pretopology of envelopes. Voevodsky shows that the h-pretopology is generated like this Zariski and the pretopology whose covers are jointly surjective families of proper morphisms. The qfh pretopology is generated like this by étale and the pretopology whose covers are jointly surjective families of finite morphisms.

Additionally, pretopologies in AG are usually superextensive (see nLab), defined via singleton pretopologies - where covering families consist of a single map. E.g. {U_i --> X} is a cov. family in etale site if \coprod U_i --> X is surjective and an etale map.
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David RobertsJul 30 '12 at 7:47

Also, the category of sheaves can be defined by the coverage in the sense of Johnstone which is just the union of the two pretopologies, rather than the pretopology which is generated by the union of the two pretopologies.
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David RobertsJul 30 '12 at 7:48

@name: your definition of an fppf cover in (C) is incorrect. There is no "quasi-finiteness" condition. And your description of fpqc is also incorrect (it entails no finite presentation condition, but one must nonetheless include a quasi-compactness condition). Finally, 17.6.2 in EGA IV is all about etale morphisms and so has no relevance to the definitions of the fppf or fpqc topologies. Perhaps you meant to say something else with (C)?
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user22479Jul 30 '12 at 14:32

@quasi-coherent: the finite presentation condition for fpqc is "inherited" from the fppf definition in the way I tried to word it. I apologise for the EGA reference - it should be 17.16.2 in the section "Quasi-sections de morphismes plats ou lisses".
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nameJul 31 '12 at 0:59

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@name: the Wikipedia article correctly defines the fpqc topology, and it involves no "finite presentation" condition (whereas you are requiring it in your item (C)). The Wikipedia article also indirectly notes that the two definitions of "fppf" (either including the quasi-finite condition or not) are equivalent (for which you have noted the right EGA reference). I have made a small change at the Wikipedia page to note that omission of quasi-finiteness is the "usual" definition of fppf, and to record the EGA reference. The definition of fpqc is rather different from what you're calling fpqc.
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user22479Aug 2 '12 at 2:45

I think of pretopologies (and coverages and other variations of this idea) as convenient ways of presenting a topology, just as groups (and other algebraic structures) can be conveniently presented by generators and relations, or (since I'm a set theorist) a complete Boolean algebra can be conveniently presented by giving a dense subposet (called a notion of forcing). Usually (though not always), the important thing is the bigger, more abstract object --- the Grothendieck topology, the group, or the complete Boolean algebra --- on which the interesting constructions can be based. Then the choice of a particular presentation is, in principle, irrelevant, but it can be very useful for working with the object and understanding it, especially because the presentation is often much smaller than the abstract object.

A Grothendieck topology by definition consists of sieves – what Johnstone calls a sifted coverage – whereas a Grothendieck pretopology in any non-trivial case will contain a non-sieve. (Recall that $\lbrace \textrm{id} : X \to X \rbrace$ is always a covering family for $X$, but it is a sieve if and only if there are no morphisms $Y \to X$ for any $Y \ne X$.) Thus, in any case of interest, no topology is a pretopology and no pretopology is a topology.

But siftedness is not the key difference between topologies and pretopologies. The key difference is saturation: as you are already aware, it is possible to add covering families to a pretopology without changing the category of sheaves. One can define the non-sifted analogue of a topology as a family of sinks satisfying the following axioms:

Any isomorphism constitutes a singleton covering family.

The composition of covering families is a covering family.

If a covering family factors through a given sink, then the sink is also a covering family.

One can show that every pretopology is contained in a unique such saturated coverage, and every saturated coverage contains a unique topology – just pick out the sieves!

Here's a reasonably "natural" example of a pair of pretopologies that generate the same topology. We consider the category $\textbf{Top}$ of topological spaces, or any full subcategory $\mathbf{T}$ thereof closed under pullbacks and open subsets.

A sink $\lbrace f_i : U_i \to X \rbrace$ is covering if and only if each $f_i$ is open and a homeomorphism onto its image, and the union of the images is the whole of $X$.

A sink $\lbrace f_i : Y_i \to X \rbrace$ is covering if and only if the induced map $f : \coprod_i Y_i \to X$ is a local homeomorphism.

Clearly, pretopology (1) is contained in pretopology (2). Conversely, given a covering family of type (2), we can obtain a covering family of type (1) by taking a suitable refinement. (For each point $y$ of $Y_i$, take an open neighbourhood $U_{i,j}$ that is mapped homeomorphically into $X$.) So the two pretopologies must generate the same topology.

This answer is useful, but I'm a bit confused by your discussion of saturation. I to indeed know and understand that Grothendieck topologies are saturated in the sense that one cannot add sieves to them without it showing up in the category of sheaves, but it's unclear to me what the axioms of a Grothendieck topology add to the axioms for a Grothendieck pretopology that guarantee this. If this comment is unclear, I can prehaps make it better when more sober in the morning.
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user25415Jul 31 '12 at 1:41

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It depends on the axioms you are thinking of. For example, the axiom that says "if you have a sieve $\mathfrak{U}$ on $X$ and a covering sieve $\mathfrak{V}$ on $X$ such that the pullback of $\mathfrak{U}$ along every morphism in $\mathfrak{V}$ is a covering sieve, then $\mathfrak{U}$ is also a covering sieve on $X$" is a saturation axiom and corresponds to the third bullet above, even though it's usually billed as a "transitivity" axiom corresponding to the second bullet!
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Zhen LinJul 31 '12 at 4:27

(I just realised my initial formulation of the third bullet was backwards. This has been fixed.)
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Zhen LinJul 31 '12 at 4:35

I think the first one is only equivalent to the other three if you add the extensive topology to those.
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Mike ShulmanAug 2 '12 at 22:52

Hmm, I guess you are right. But don't they all give the same category of sheaves? There might be a span of sites connecting the first of the above to each of the others.
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David RobertsAug 3 '12 at 0:36

Nope. A sheaf on the open cover site must be trivial over the empty manifold; the other ones don't even force that.
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Mike ShulmanAug 3 '12 at 5:25

Hmm, you're right. I'll take out the first one.
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David RobertsAug 3 '12 at 10:05

More generally, a topology -- as opposed to a pretopology -- is uniquely determined by its category of sheaves (as a subcategory of the category of presheaves).
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Mike ShulmanAug 3 '12 at 18:04